+128474 13^n - 6^(n - 2)
Show it's true for n = 2
13^2 - 6^(2 - 2) =
169 - 1 = 168
168 / 7 = 24
Assume that this is true for n = k where k ≥ 2
That is
13^k - 6^(k - 2) is divisible by 7
Prove it's true for k + 1
That is
13^(k + 1) - 6^ (k + 1 - 2) is divisible by 7
[ note ....6^(k + 1 - 2) = 6^(k - 2 + 1) ]
So we have
13^(k+ 1) - 6 ^( k - 2 + 1)
13^k * 13^1 - 6^(k-2) * 6^1
13 * 13^k - 6 * 6^(k - 2)
(6 + 7) 13^k - 6 * 6^(k - 2)
6 [ 13^k - 6^(k - 2) ] + 7 * 13^k
And since we assumed that 13^k - 6^(k - 2) was divisible by 7, then the first term is divisible by 7, as well
And the second term, 7*13^k, is divisible by 7
